8. powder mixing and blending; fluid flow through …...fig_mpe_8 dr. w. hintz + prof. dr. tomas...

Fig. 8.1

Prof. Dr. J. Tomas, chair of Mechanical Process Engineering

Fig_MPE_8 Dr. W. Hintz + Prof. Dr. Tomas Mechanical Process Engineering - Particle Technology Powder mixing and blending 30.06.2014 Figure 8.1

8. Powder mixing and blending;

Fluid flow through

particle beds

8. Product tailoring by particle mixing,

8.1 Microprocesses and mixing efficiency of particles 8.1.1 Model of stochastic homogeneity 8.1.2 Mixing kinetics

8.2 Rotating vessels, kneaders and agitators 8.3 Pneumatic mixing

8.3.1 Permeation of particle beds 8.3.2 Fluidized bed mixer

Fig. 8.2



Processes or unit operations of mechanical process engineering according to RUMPF

separation

combination

without

change

of

particle size

mechanical

separation (filters,

separators,

screens, sifters)

powder mixing

and

blending

particle

size

analysis with change

of

particle size

size reduction

(crushing and

grinding)

size enlargement

(agglomeration)

transport and storage of bulk materials

Fig. 8.3



Characterisation of particle mixtures

1. characterisation of mixing states of a granular material or

particle system

a) completely separated

b) ideal mixture

(regular distribution)

c) random mixture

c)

Fig. 8.4



2. Characterisation of the composition of a mixture and its

quality of mixing

a) expected value µp = mp/m of a composition of a mixture is

known

( )∑=

µ−µ⋅=n

1j

2pj,p

2

n1s~

s~ standard deviation, 2s~ estimated variance of the expected value

n number of samples

µp,j mass fraction of the component p in the jth sample

a) expected value of a composition of a mixture is not known and

has to be estimated

∑=

µ⋅=µn

1jj,pp n

1 mean value (estimated expectation of µp)

( )∑=

µ−µ⋅−

=n

1j

2pj,p

2

1n1s~

3. Variance σZ2 of stochastic homogeneity or an ideal random

mixture, consisting of two components according to STANGE

( ) ( )[ ]2ppq

2qqp

S

qp2Z v1mv1m

m+⋅⋅µ++⋅⋅µ⋅

µ⋅µ=σ

µp, µq mass fractions of the components (P) and (Q) mS mass of a single sample

qp m,m mean particle mass of the components (P) and (Q)

Fig. 8.5



q

qq

p

pp m

v,m

vσ

=σ

= variation coefficients of the particle mass (size)

distributions of the components (P) and (Q) 4. Special cases:

a) narrow size distributions of both components vp → 0, vq → 0,

average particle number in one sample pSp m/mN =

pqp2Z N/µ⋅µ≈σ

b) nearly equivalent size distributions qp mm ≈ , vp ≈ vq

( )2qq

S

qp2Z v1m

m+⋅⋅

µ⋅µ≈σ

c) large mass fraction of component (P) µp > 0.9 and qp mm ≈

( )2qq

S

qp2Z v1m

m+⋅⋅

µ⋅µ≈σ

d) component (P) is very coarse qp mm >>

( )2pp

S

qp2Z v1m

m+⋅⋅

µ⋅µ≈σ

5. Mixing efficiency ME = 1 – SE segregation index:

s~/ME Z1 σ= ⇒ ( ) 1...0...s~/ ZmaxZ =σσ=σ

( ) qppp2max 1 µ⋅µ=µ−⋅µ=σ

max3 /s~1ME σ−= ⇒ ( ) 1...0/...s~1 maxZmax =σσσ=−

Fig. 8.6



Process principles for input of mechanical energy into particulate systems

Process principle schematic sketch example

rotating process

chambers or vessels

drum mixer drum mill

drum dryer

agitated systems

mechanical silo

discharging forced mixing

impact mill

flow of fluids

through granular

beds

pneumatic silo

discharging pneumatic

homogenisation fluidised beds

vibrations

vibration systems for charging and

discharging screen machine

vibration mill

Fig. 8.7



Mixing - of granular materials 1. drum mixer

a) cylindrical mixing drum d) cubic mixer

b) double cone mixer e) tetrahedral mixer

c) double cone mixer with a tumbling f ) V - mixer

Fig. 8.8



Motion processes in a drum mixer

a) small number of revolutions b) higher number of revolutions

c) number of revolutions near ncrit.

cascade zone cascade zone cascade zone

cataract zone

Fig. 8.9



Mixing processes in a drum mixer

a) before mixing b) after mixing c) composition of samples taken from different

heights of the mixing bulk materials, before mixing

(t = 0), while mixing, and after mixing (t > tM)

Fig. 8.10



s2 = f (t) for different mixing processes a) without segregation processes

b) with segregation processes in the

beginning, components of higher

density are above those of lower

density

c) with segregation processes in the

beginning, components of higher

density are under those of lower

density

Fig. 8.11



Mixing - of granular materials

2. kneader, pan mixer

a) double-shaft kneader or pan mixer b) fast running paddle mixer

1 rotating mixing tool A feed

2 repel share, ploughshare M mixed product

Fig. 8.12



Mixing - of granular materials

3. Pneumatic mixer

a) fluidised bed mixer b) air jet mixer

LE air inlet

LA air outlet

A feed material for mixing

S1 – S4 separate aeration segments

Fig. 8.13



Fig. 8.14



Flow of fluids through granular beds - important mechanical

micro processes

1. schematic flow of fluids through a granular bed

entrance streaming profile of fluid exit streaming profile of fluid P2 - P1 pressure drop across the bed

u average velocity of flow of the fluid

uε average velocity of flow of the fluid in the pores

ε porosity of the bed

A total cross sectional area of the bed

I thickness of the bed

Fig. 8.15



2. different states of granular beds while fluid permeation

bulk material begin of homogeneous inhomogeneous unsteady

fluidisation fluidised bed fluidised bed fluidised bed

gas or liquid gas or liquid liquid gas gas gas or liquid

low velocity of flow high velocity of flow

of the fluid of the fluid

Fig. 8.16



3. pressure drop in the granular bed dependent on the fluid flow rate

Fig. 8.17



Fluid Flow through Particle Beds – Permeation Models a) Fixed particle bed

Valid for Model of Remarks

No Author Equation

Lam

inar

flow

-aro

und

Re

< 0.

5 tr

ansi

tion

Re

= 0.

5 - 1

000

Tur

bule

nt fl

ow-a

roun

d

Re

> 10

00

Poro

sity

ε

Pore

syst

em

Ran

dom

pac

king

mon

odis

pers

e sp

here

s

Part

icle

shap

e

1 Darcy uKukh

p

B⋅η⋅=⋅=

∆, K = Darcy constant

+ - - - + - - - Homoge-neous pore system

2 Carman, Kozeny

( )CK

2V,S3

2

B

KuA1h

p⋅⋅η⋅⋅

εε−

=∆

, KCK = Car-

man Kozeny constant, KCK = 5 for spheres of equal size with small deviation

+ - -

≈ 0.

4

+ - + - parallel bended, cylindrical pores

3 Gupte 5.5

B

K2

f Re6.5

hd

up

ε⋅=⋅

⋅ρ∆

+ - - + - + + - Dimension

analysis 4 Molerus,

Pahl, Rumpf

( ) 55.4B

K2

f Re6.514

hd

up

ε⋅⋅ε−⋅=⋅

⋅ρ∆

+ - -

0.35

– 0

.7 - + + - Basing on

results of Gupte

Fig. 8.18



5 Pärnt ( )3

2

2ST

2R

2F

hB

1d

uReh

pεε−

⋅⋅ψ⋅ψ

⋅η⋅⋅ξ=

∆,

ε−=

1ReReh

Reh = hydraulic Reynolds number ψF = shape factor, ψR = roundness factor ξ = fluid drag coefficient

+ - - + - + - + Experiments with fine disperse particle beds

6 Burke, Plummer 3

B

K2

f

175.1hd

up

εε−

⋅=⋅⋅ρ∆

- - + + - - - - Experiments

7 Ergun ( )3

B

K2

f

1hd

up

εε−

⋅λ=⋅⋅ρ∆

, 75.1Re

1150 +ε−

⋅=λ + + + + - - - - Experiments

8 Molerus

1.095.0

5.1

95.0

95.0

2

B

Re891.0

ad4.0

ad12.01

Re4

ad

21

ad692.01

Re24Eu

⋅

++

⋅+⋅+

+

⋅+⋅+⋅=

Euler number of fixed bed:

ε−ε

⋅ρ∆⋅

=1h

du3p4Eu

2

B2

fB

Re < 104

0.1

- 1

+ + + + η

ρ⋅⋅ε= fd)/u(

Re

Surface separation ratio

3

3

95.0 195.01

ad

ε−−ε−

=

d+a

d

a

d

d+a

Fig. 8.19



b) Fluidized bed Valid for Model of

Remarks

No Author Equation

Lam

inar

flow

-aro

und

Re

< 0.

5 T

rans

ition

R

e =

0.5

- 100

0 T

urbu

lent

flow

-aro

und

R

e >

1000

Po

rosi

ty ε

Pore

syst

em

Ran

dom

pac

king

mon

odis

pers

e sp

here

s Pa

rtic

le sh

ape

9 Beranek, Rose, Winter-stein

( ) ( ) g1h

pfsL

B⋅ρ−ρ⋅ε−=

∆

εL = Porosity at point of bed expansion

- - + + + + + - ∆p = const. for complete range of fluidized bed

10 Molerus

1.0

5.1

2

W

Re907.0

ad4.0

ad07.01

Re4

ad

21

ad341.01

Re24Eu

⋅++

⋅+⋅+

+

⋅+⋅+⋅=

Euler number of fluidized bed:

2f

fsW )/u(

gd34Eu

ε⋅

⋅ρρ−ρ

⋅= with WW c

1Eulim

=→ε

Re < 104

0.5

- 1

+ + + + η

ρ⋅⋅ε= fd)/u(

Re

Surface separation ratio

3

3

9.0 19.01

ad

ε−−

ε−=

d

d+a d+a a

d

Fig. 8.20



Fig. 8.21



Survey of constitutive functions, processing and handling problems of cohesive powders. Property, problems

Physical principle

Physical assessment of product quality Particle size

d in µm Physical law Assessment

characteristic Value range

Evaluation

Large adhesion potential1) FG

FH0

2s

20

sls,H

G

0H

dag2C

FF

⋅ρ⋅⋅π= 2

2

d)µm100(

WeightAdhesion

≈ 1 - 100

100 - 104 104 - 108

slightly adhesive adhesive

very adhesive

10 - 100 1 - 10

0.01 - 1 Large intensifi-cation of adhesion2)

FN

FH(FN)FN

FH(FN)

pA

p

κ−κκ

=κ Contact consolidation

coefficient κ by flattening

0.1 – 0.3 0.3 – 0.77

> 0.77

soft very soft

extreme soft

< 10 < 1

< 0.1

Poor flowablity2)

σ1 σc

c

1cff

σσ

= Flow function ffc 2 - 4

1 - 2 < 1

cohesive very cohesive non-flowing

< 100 < 10 < 0.1

Large compres-sibility2)

σ1

∆h

n

0

st,M

0,b

b 1

σσ

+=ρρ

Compressibility index

n 0.05 – 0.1

0.1 - 1 compressible

very compressible < 100 < 10

Small permea-bility3,4)

∆hW

∆hb

u b

Wf h

hku∆∆⋅=

Permeability kf in m/s

< 10-9 10-9 - 10-7 10-7 - 10-5

non-permeable very low

low

< 1 1 - 10

10 - 100

Poor fluidi-sation5,6)

( ))d(ufp P=∆ Channelling Group C, non-fluidising

< 10

1) Rumpf, H.: Die Wissenschaft des Agglomerierens. Chem.-Ing.-Technik, 46 (1974) 1-11. 2) Tomas, J.: Product Design of Cohesive Powders - Mechanical Properties, Compression and Flow Behavior. Chem. Engng. & Techn., 27 (2004) 605-618. 3)Förster, W.: Bodenmechanik - Mechanische Eigenschaften der Lockergesteine, 4. Lehrbrief, Bergakademie Freiberg 1986. 4) Terzaghi, K., Peck, R. B., Mesri, G.: Soil mechanics in engineering practice, Wiley, New York 1996.

5) Geldart, D.: Types of Gas Fluidization, Powder Techn. 7 (1973) 285-292. 6) Molerus, O.: Fluid-Feststoff-Strömungen, Springer, Heidelberg 1982.

8. powder mixing and blending; fluid flow through …...fig_mpe_8 dr. w. hintz + prof. dr. tomas...

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